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Cayley's Sextic

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CayleysSextic

A plane curve discovered by Maclaurin but first studied in detail by Cayley. The name Cayley's sextic is due to R. C. Archibald, who attempted to classify curves in a paper published in Strasbourg in 1900 (MacTutor Archive). Cayley's sextic is given in polar coordinates by

r=4acos^3(1/3theta).
(1)

The Cartesian equation is

-a^3x^3-48ax(x^2+y^2)^2+64(x^2+y^2)^3 -3a^2(x^2+y^2)(5x^2+9y^2)=0.
(2)

Parametric equations can be given by

x(t)=4acos^3(1/3t)cost
(3)
y(t)=4acos^3(1/3t)sint
(4)

for 0<t<3pi. In this parametrization, the loop corresponds to pi<t<2pi.

The area enclosed by the outer boundary is

A=(5pi+9/2sqrt(3))a
(5)
=23.50219...a
(6)

(OEIS A118308), and by the inner loop is

A_(loop)=1/2(5pi-9sqrt(3))a^2
(7)
=0.05975299a^2...
(8)

(OEIS A118309), and the arc length of the entire curve is

s=6pia.
(9)

The arc length, curvature, and tangential angle are given by

s(t)=[2t+3sin(2/3t)]a
(10)
kappa(t)=(4sec^2(1/3t))/(3a),
(11)
phi(t)=4/3t.
(12)

See also

Cayley's Sextic Evolute

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References

Gray, A. Modern Differential Geometry of Curves and Surfaces with Mathematica, 2nd ed. Boca Raton, FL: CRC Press, pp. 119-120, 1997.Lawrence, J. D. A Catalog of Special Plane Curves. New York: Dover, pp. 178 and 180, 1972.MacTutor History of Mathematics Archive. "Cayley's Sextic." http://www-groups.dcs.st-and.ac.uk/~history/Curves/Cayleys.html.Sloane, N. J. A. Sequences A118308 and A118309 in "The On-Line Encyclopedia of Integer Sequences."

Cite this as:

Weisstein, Eric W. "Cayley's Sextic." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CayleysSextic.html

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